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2007 AMC 12A Problem 15

Problem 15 of 25IntermediateAlgebra

The set {3,6,9,10}\{3,6,9,10\} is augmented by a fifth element n,n, not equal to any of the other four. The median of the resulting set is equal to its mean. What is the sum of all possible values of n?n?

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Solution

The mean is 28+n5.\dfrac{28+n}{5}. If n<6,n\lt 6, the median is 6,6, so 28+n=3028+n=30 and n=2.n=2. If 6<n<9,6\lt n\lt 9, the median is n,n, so 28+n=5n28+n=5n and n=7.n=7. If n>9,n\gt 9, the median is 9,9, so 28+n=4528+n=45 and n=17.n=17. The sum of all possible values is 2+7+17=26.2+7+17=26. Thus, the correct answer is E.

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Concepts: median (data) · mean · casework

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.